Skip to main content
SpringerLink
Log in

Necessary and Sufficient Conditions for the Existence of a Classical Solution of the Mixed Problem for the Homogeneous Wave Equation with an Integrable Potential

  • Partial Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We use the Fourier method to obtain necessary and sufficient conditions for the existence of a classical solution of the mixed problem for a homogeneous wave equation with an integrable potential and fixed endpoints. The solution is represented by a rapidly convergent series.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Khromov, A.P., On the convergence of the formal Fourier solution of the wave equation with a summable potential, Comput. Math. Math. Phys., 2016, vol. 56, @@no. 10, pp. 1778–1792.

    Article  MathSciNet  MATH  Google Scholar 

  2. Kornev, V.V. and Khromov, A.P., On a generalized formal Fourier solution of the mixed problem for an inhomogeneous wave equation, in Mater. 19-i mezhdunar. Saratovskoi zimnei shkoly “Sovremennye problemy teorii funktsii i ikh priblizhenii,” posvyashch. 90-letiyu akad. P.L. Ul’yanova (Proc. 19th Int. Saratov Winter Sch. “Modern Problems of the Theory of Functions and Their Approximations,” Dedicated 90th Anniv. Acad. P.L. Ul’yanov), January 29–February 2, Saratov, 2018, pp. 156–159.

    Google Scholar 

  3. Kornev, V.V. and Khromov, A.P., On solving an inhomogeneous wave equation with fixed end points and zero initial conditions, in Mater. 19-i mezhdunar. Saratovskoi zimnei shkoly “Sovremennye problemy teorii funktsii i ikh priblizhenii,” posvyashch. 90-letiyu akad. P.L. Ul’yanova (Proc. 19th Int. Saratov Winter Sch. “Modern Problems of the Theory of Functions and Their Approximations,” Dedicated 90th Anniv. Acad. P.L. Ul’yanov), January 29–February 2, Saratov, 2018, pp. 159–160.

    Google Scholar 

  4. Kornev, V.V. and Khromov, A.P., On the classical and generalized solution of the mixed problem for the wave equation, in Mater. mezhdunar. konf. “Sovremennye metody teorii krayevykh zadach. Pontryaginskie chteniya—XXIX,” posvyashch. 90-letiyu akad. V.A. Il’ina (Proc. Int. Conf. “Modern Methods of the Theory of Boundary Value Problems. Pontryagin Readings—XXIX,” Dedicated 90th Anniv. Acad. V.A. Il’in), May 2–6, 2018, Moscow, 2018, pp. 132–133.

    Google Scholar 

  5. Rasulov, M.L., Metod konturnogo integrala i ego primenenie k issledovaniyu zadach dlya differentsial’nykh uravnenii (The Contour Integral Method and Its Application to Problems for Differential Equations), Moscow: Nauka, 1964.

    Google Scholar 

  6. Vagabov, A.I., Vvedenie v spektral’nuyu teoriyu differentsial’nykh operatorov (Introduction to Spectral Theory of Differential Operators), Rostov-on-Don: Rostov State Univ., 1994.

    Google Scholar 

  7. Levitan, B.M., Operatory obobshchennogo sdviga i nekotorye ikh primeneniya (Generalized Shift Operators and Some of Their Applications), Moscow: Gos. Izd. Fiz.-Mat. Lit., 1962.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chernyshevsky Saratov State University, Saratov, 410012, Russia

    A. P. Khromov

Authors
  1. A. P. Khromov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. P. Khromov.

Additional information

Russian Text © The Author(s), 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 5, pp. 717–731.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khromov, A.P. Necessary and Sufficient Conditions for the Existence of a Classical Solution of the Mixed Problem for the Homogeneous Wave Equation with an Integrable Potential. Diff Equat 55, 703–717 (2019). https://doi.org/10.1134/S0012266119050112

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266119050112

Search

Navigation